Of course she has to be registered to pay Tax and NI. If she worked for you cash in hand and claimed benefits, she is breaking the law and could face a custodial sentence.
If she doesn't claim benefits and only has what income you pay her, she looses her NI which will affect her state pension.
She either needs to register as a childminder in which case you would be paying the going rate any way, or you employ her as a Nanny which will be above going rate (I get £23 per hour for example for two children) but either way, you would have to pay her minimum wage.
She can't be self employed if she is only nannying for one family, however if she nannies for more than one family, she can be self employed.
Registering as a Childminder isn't a walk in the park. It involves Ofsted, house inspections, health and safety, safeguarding etc. It's not a short process.
Childcare is expensive, no two ways about it but surely it's worth paying knowing your child is being looked after in the best possible way and by a reliable and experienced child care provider?